Competition · AMC preparation · step 4 of 4
AMC 8 · 2002 · #1
Grade 7 geometry-2dcountingPick an answer.
AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Three figures make three pairs: line-line, line-circle (twice). Tool #7 (Break into Subproblems) lets us handle one pair at a time, find the maximum for each, and add. Tool #15 (Visualize) is needed at the end to draw a single picture where every pair hits its maximum at the same time, so the five points are all different and the total is actually reachable.
Cross the two lines
Pair 1, line vs line: two distinct lines cross at most once, so cross them for the max of 1 point.
Grade 4 introduces points, lines, and parallel vs. intersecting lines: two different lines share at most one point.
4.G.A.1Identify SubproblemsCross the circle and first line
Pair 2, circle vs first line: a secant through the inside cuts the circle at the max of 2 points.
Grade 7 work with circles makes the secant case familiar: a chord meets the circle at its two endpoints.
7.G.B.4Identify SubproblemsCross the circle and second line
Pair 3, circle vs second line: independent of the first, it is also a secant for another 2 points.
The second line is independent of the first, so it can also be a secant and contribute 2 more points.
7.G.B.4Identify SubproblemsAdd the three maximums
Add the three pair-maximums, 1 + 2 + 2, to bound the total at 5.
Grade 4 multi-step addition: each pair contributes independently, so the totals just stack up.
The most crossing points possible is the two lines' single crossing added to each line's pair of crossings with the circle, with every one of those points kept distinct.
▸ Why?
The crossing points split into three separate pair-groups — the two lines with each other, and the circle with each line — and a drawing can keep these groups from sharing any point, so the overall count is exactly the three group-counts added together.
▸ Why?
When a collection is cut into parts that overlap nowhere and leave nothing out, the size of the whole equals the part-sizes added up.
▸ Why?
A single straight line can cross the circle in at most two places, so each line adds at most two points to the total.
▸ Why?
Every point of the circle lies one fixed radius from the center, so a line's shared points sit balanced in a pair about its closest approach to the center, and no third shared point can fit.
Check the bound is reachable
Sketch a circle with two secants crossing inside it: 1 inner crossing + 4 on the circle = 5 distinct points, so the bound is reached.
Grade 4 "draw and identify" geometry: a quick sketch confirms the five points are distinct, so the upper bound 5 is actually attained.
4.G.A.1Organize Information In More WaysThree figures make three pairs. Find each pair's biggest intersection count, add them up, and then make sure one picture can hit all the maximums at once — that final check is what turns 5 from a guess into the real answer.
- Cross the two lines
- Cross the circle and first line
- Cross the circle and second line
- Add the three maximums
- Check the bound is reachable
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